Infinitely many solutions of a semilinear problem for the Heisenberg Laplacian on the Heisenberg group

We study the semilinear equationwhere is the Heisenberg Laplacian and is the Heisenberg group. The function f C²(×, ) is supposed to satisfy some (subcritical) growth conditions and to be left invariant under the action of the subgroup of consisting of points with integer coordinates.. We show the existence of infinitely many solutions in the space S₁²(), which is the Heisenberg analogue of the Sobolev space W^1,2(^N).Mathematics Subject Classification (2000): 22E30, 22E27     

Heterometallic metal-organic frameworks based on tris(dipyrrinato) coordination complexes

A novel class of heterometallic metal-organic frameworks (MOFs) has been synthesized and characterized. The MOFs rely on the use of tris(dipyrrinato) coordination complexes as the bridging structure and silver(I) ions as the linking unit. The building blocks and resulting MOFs have been structurally characterized by using single-crystal X-ray diffraction. The modular nature of this approach is demonstrated by the use of both iron(III) and cobalt(III) complexes. The MOFs have strong electronic absorption features originating from the metal-dipyrrin chromophore and have continuous channels throughout the lattice that are occupied by ordered and disordered solvent molecules.     

Rate constants for the reaction of OH radicals with a series of alkanes and alkenes at 299 ± 2 K

Using methyl nitrite photolysis in air as a source of hydroxyl radicals, relative rate constants for the reaction of OH radicals with a series of alkanes and alkenes have been determined at 299 ± 2 K. The rate constant ratios obtained are: relative to n-hexane = 1.00, neopentane 0.135 ± 0.007, n-butane 0.453 ± 0.007, cyclohexane 1.32 ± 0.04; relative to cyclohexane = 1.00, n-butane 0.341 ± 0.002, cyclopentane 0.704 ± 0.007, 2,3-dimethylbutane 0.827 ± 0.004, ethene 1.12 ± 0.05; relative to propene = 1.00, 2-methyl-2-butene 3.43 ± 0.13, isoprene 3.81 ± 0.17, 2,3-dimethyl-2-butene 4.28 ± 0.21. These relative rate constants are placed on an absolute basis using previous absolute rate constant data and are compared and discussed with literature data.     

Kinetics of the gas-phase reactions of OH radicals with alkyl nitrates at 299 ± 2 K

Relative rate constants for the gas-phase reactions of OH radicals with a series of alkyl nitrates have been determined at 299 ± 2 K, using methyl nitrite photolysis in air as a source of OH radicals. Using a rate constant for the reaction of OH radicals with cyclohexane of 7.57 × 10^?12 cm³/molec·s, the rate constants obtained are (× 10¹² cm³/molec·s): 2-propyl nitrate, 0.18 ± 0.05; 1-butyl nitrate, 1.42 ± 0.11; 2-butyl nitrate, 0.69 ± 0.10; 2-pentyl nitrate, 1.87 ± 0.12; 3-pentyl nitrate, 1.13 ± 0.20; 2-hexyl nitrate, 3.19 ± 0.16; 3-hexyl nitrate, 2.72 ± 0.22; 3-heptyl nitrate, 3.72 ± 0.43; and 3-octyl nitrate, 3.91 ± 0.80. These rate constants, which are the first reported for the alkyl nitrates, are significantly lower than those for the parent alkanes, and a formula, based on the numbers of the various types of C? H bonds in the alkyl nitrates, is derived for rate constant estimation purposes.     

NH<Subscript>3</Subscript>(CH<Subscript>2</Subscript>)<Subscript>5</Subscript>NH<Subscript>3</Subscript>BiCl<Subscript>5</Subscript>: an efficient and recyclable catalyst for the synthesis of benzoxazole,benzimidazole and benzothiazole heterocycles

In this work, the condensation of aromatic aldehydes with different two-substituted aniline catalyzed by NH₃(CH₂)₅NH₃BiCl₅ as heterogeneous and recyclable catalyst was reported. It was demonstrated that NH₃(CH₂)₅NH₃BiCl₅ can act as an efficient and active catalyst and is reusable for six runs without a significant loss of their catalytic activity. Simple preparation of the catalyst, high catalytic activity and good reusability are noteworthy advantages of this catalytic system in the synthesis of benzoxazole, benzimidazole and benzothiazole heterocycles at room temperature under solvent-free conditions.     

MnCo<Subscript>2</Subscript>O<Subscript>4</Subscript> nanoparticles with excellent catalytic activity in thermal decomposition of ammonium perchlorate

MnCo₂O₄ spinel nanoparticles (NPs) have been prepared using Aloe vera gel solution. The characterization of prepared spinel was performed applying Fourier transform infrared spectroscopy, X-ray diffraction, Raman spectroscopy, transmission electron spectroscope, scanning electron microscope and dynamic light scattering. The results manifested that the prepared nanoparticles were mainly spherical plus minor agglomeration with average size distribution between 35 and 60 nm. The catalytic activity of the prepared nanoparticles upon thermal degradation of ammonium perchlorate (AP) was evaluated applying differential scanning calorimetry and thermogravimetry instruments. MnCo₂O₄ nanoparticles increased the released heat of AP from 450 to 1480 J g^?1 and decreased the decomposition temperature from 420 to 293 °C. The kinetic parameters obtained from Kissinger methods showed that the activation energy of AP thermal decomposition in the presence of MnCo₂O₄ NPs considerably decreased. Also, a mechanism has been proposed in the presence of catalyst for the process of thermal decomposition of AP.     

Iron‐Catalyzed Reductive Ethylation of Imines with Ethanol

The borrowing hydrogen strategy has been applied to the ethylation of imines with an air‐stable iron complex as precatalyst. This approach opens new perspectives in this area as it enables the synthesis of unsymmetric tertiary amines from readily available substrates and ethanol as a C₂ building block. A variety of imines bearing electron‐rich aryl or alkyl groups at the nitrogen atom could be efficiently reductively alkylated without the need for molecular hydrogen. The mechanism of this reaction, which shows complete selectivity for ethanol over other alcohols, has been studied experimentally and by means of DFT computations.     

Outstanding Methane Oxidation Performance of Palladium‐Embedded Ceria Catalysts Prepared by a One‐Step Dry Ball‐Milling Method

By carefully mixing Pd metal nanoparticles with CeO₂ polycrystalline powder under dry conditions, an unpredicted arrangement of the Pd‐O‐Ce interface is obtained in which an amorphous shell containing palladium species dissolved in ceria is covering a core of CeO₂ particles. The robust contact that is generated at the nanoscale, along with mechanical forces generated during mixing, promotes the redox exchange between Pd and CeO₂ and creates highly reactive and stable sites constituted by PdO_x embedded into CeO₂ surface layers. This specific arrangement outperforms conventional Pd/CeO₂ reference catalysts in methane oxidation by lowering light‐off temperature by more than 50°C and boosting the reaction rate. The origin of the outstanding activity is traced to the structural properties of the interface, modified at the nanoscale by mechanochemical interaction.     

A simple extrapolation method for clustered eigenvalues

Numerical Algorithms - This paper introduces a simple variant of the power method. It is shown analytically and numerically to accelerate convergence to the dominant eigenvalue/eigenvector pair,...     

Products of the gas-phase reaction of methyl tert-butyl ether with the OH radical in the presence of NOx

The products of the reaction of the hydroxyl (OH) radical with methyl tert-butyl ether (MTBE) in NO_x-air systems were identified and measured by Fourier transform infrared absorption spectroscopy and gas chromatography. The products observed, and their yields, were as follows: t-butyl formate, 76 ± 7%; formaldehyde, 37%; methyl acetate, 17 ± 2%, and acetone, 2.1 ± 0.9%, where the stated error limits represent both random (two standard deviations) and estimated systematic uncertainties. These products account for ca. 95% of the MTBE carbon reacted. Infrared absorption bands which may be due to small amounts of organic nitrate formation were observed, but organic nitrate yields could not be quantified. These data allow a chemical mechanism for the reaction of MTBE with the OH radical in the presence of NO_x to be formulated.