Spectral theory approach for a class of radial indefinite variational problems

Considering the radial nonlinear Schrödinger equation

($P_r$)$?\mathrm{\Delta }u+V(x)u=g(x,u)\text{in}{\mathbb{R}}^N,N\ge 3$

we aim to find a radial nontrivial solution for it, where V changes sign ensuring problem ($P_r$) is indefinite and g is an asymptotically linear nonlinearity. We work with variational methods associating problem ($P_r$) to an indefinite functional in order to apply our Abstract Linking Theorem for Cerami sequences in [8] to get a non-trivial critical point for this functional. Our goal is to make use of spectral properties of operator $A:=\mathrm{\Delta }+V(x)$ restricted to $H_{rad}^1({\mathbb{R}}^N)$, the space of radially symmetric functions in $H^1({\mathbb{R}}^N)$, for obtaining a linking geometry structure to the problem and by means of special properties of radially symmetric functions get the necessary compactness.     

New lower bounds on the radius of spatial analyticity for the KdV equation

The radius of spatial analyticity for solutions of the KdV equation is studied. It is shown that the analyticity radius does not decay faster than $t^{?1/4}$ as time t goes to infinity. This improves the works of Selberg and da Silva (2017) [30] and Tesfahun (2017) [34]. Our strategy mainly relies on a higher order almost conservation law in Gevrey spaces, which is inspired by the I-method.     

Identification and quantification of main flavonoids in the leaves of Bambusa multiplex cv. Fernleaf

Abstract

The chemical profile of Bambusa multiplex cv. Fernleaf (B. multiplex) leaves was analysed by UPLC-DAD-Q-TOF-MS. Twelve compounds were identified and C-glycosyl flavonoids, including vitexin, isovitexin, isoorientin and its derivatives, are the main constitutes of the plant. Besides, a HPLC method for isoorientin quantification was developed. The RSD of retention time and peak area were 0.05% and 2.04% for six times analysis of isoorientin with concentration of 20?μg/mL. The recovery of isoorientin in real sample was 99.2%. The general trend of isoorientin content in B. multiplex leaves was that it steady increased from Jan. to May, and then quickly decreased. The maximum was found on May with value of 4.7?mg/g. The lowest level of isoorientin was found during Aug. to Nov. with value of about 1.66?mg/g. In different seasons, isoorientin is always the most dominant flavonoid which was accounted for about 50% of total flavonoids in the sample.     

A generalized super AKNS hierarchy associated with orthosymplectic Lie superalgebra OSP(2,2) and its super bi-Hamiltonian structures

For the orthosymplectic Lie superalgebra $\text{◂⋅▸}OSP(2,2)$, we choose a set of basis matrices. A linear combination of those basis matrices presents a spatial spectral matrix. The compatible condition of the spatial part and the corresponding temporal parts of the spectral problem leads to a generalized super AKNS (GSAKNS) hierarchy. By making use of the supertrace identity, the obtained GSAKNS hierarchy can be written as the super bi-Hamiltonian structures.     

The Performance of Jatropha Curcas Linn. Capsule Husk as Feedstocks Biogas in One Phase Anaerobic Digestion

Jatropa curcas Linn. (JcL) capsule husk was not recommended as biogas feedstocks. However for biorefinery purpose, several technologies have been conducting to solve this problem. This research reported quantity and quality comparison of Dry Husk Jcl (DH-JcL) in one phase system of batch digester compare with semi continuous digester. HDPE drum of 80 L working volume used as digester with 40 days hydraulic retention time. Feeding of DH-Jcl and solvent water was mixed on concentration of 1: 8. Research conclusion showed that semi continuous digester was better than batch digester. Biogas quality showed that methane content can reach 66.61% to 83.15% and biogas quantity in semi continuous digester can reach 0.016 m³ · kg^–1 DH JcL. The result was not in optimize condition yet because ratio number of volatile fatty acids/ alkalinity showed 0.5, it was indicated unstable anaerobic degradation process of DH-JcL.     

Unprecedented Oxidative Addition and Metal‐Only Lewis Pair Chemistry of Antimony Trihalides

Reaction of the zero‐valent platinum complex [Pt(PCy₃)₂] with SbF₃ generates the cationic diplatinum stibenium complex [{(Cy₃P)₂Pt}₂(μ‐SbF₂)]⁺, the first unsupported metal‐only Lewis pair containing an antimony‐centered Lewis acid. In contrast, SbCl₃ undergoes oxidative addition to [Pt(PCy₃)₂], resulting in the dihalostibanyl complex trans‐[PtCl(SbCl₂)(PCy₃)₂], the first example of oxidative addition of an antimony–halide bond to a transition metal.     

New fabrication approach to develop a high birefringence photo-crosslinked film based on a sulfur-containing liquid crystalline molecule with large temperature dependence of birefringence

ABSTRACT

We present a new fabrication approach to achieve a high birefringence film by means of photopolymerization based on an alkylthio-containing rod-like liquid crystalline molecule exhibiting large temperature dependence of birefringence. We designed a new reactive mesogen having alkylthio linkages (BPM–S). It was found that BPM–S had a larger increment of birefringence with decreasing temperature, relative to commercially available alkoxy analog LC242. This result could be thought to be due to enhanced intermolecular attractive interaction for an alkylthio mesogen implied by the proximity of laterally neighboring molecules and cybotactic nematic tendency based on wide-angle X-ray measurement. The uniaxially-aligned photo-polymerized film for BPM–S showed higher birefringence than that for LC 242.     

Group 6 Oxyfluoro-Anion Salts of [XeF5]+ and [Xe2F11]+: Syntheses and Structures of [XeF5][M2O2F9] (M=Mo,W), [Xe2F11][M′OF5] (M′=Cr,Mo, W), [XeF5][HF2]⋅CrOF4, and [XeF5][WOF5]⋅XeOF4

The reactions of the fluoride-ion donor, XeF₆, with the fluoride-ion acceptors, M′OF₄ (M′=Cr, Mo, W), yield [XeF₅]⁺ and [Xe₂F₁₁]⁺ salts of [M′OF₅]⁻ and [M₂O₂F₉]⁻ (M=Mo, W). Xenon hexafluoride and MOF₄ react in anhydrous hydrogen fluoride (aHF) to give equilibrium mixtures of [Xe₂F₁₁]⁺, [XeF₅]⁺, [(HF)_nF]⁻, [MOF₅]⁻, and [M₂O₂F₉]⁻ from which the title salts were crystallized. The [XeF₅][CrOF₅] and [Xe₂F₁₁][CrOF₅] salts could not be formed from mixtures of CrOF₄ and XeF₆ in aHF at low temperature (LT) owing to the low fluoride-ion affinity of CrOF₄, but yielded [XeF₅][HF₂]⋅CrOF₄ instead. In contrast, MoOF₄ and WOF₄ are sufficiently Lewis acidic to abstract F⁻ ion from [(HF)_nF]⁻ in aHF to give the [MOF₅]⁻ and [M₂O₂F₉]⁻ salts of [XeF₅]⁺ and [Xe₂F₁₁]⁺. To circumvent [(HF)_nF]⁻ formation, [Xe₂F₁₁][CrOF₅] was synthesized at LT in CF₂ClCF₂Cl solvent. The salts were characterized by LT Raman spectroscopy and LT single-crystal X-ray diffraction, which provided the first X-ray crystal structure of the [CrOF₅]⁻ anion and high-precision geometric parameters for [MOF₅]⁻ and [M₂O₂F₉]⁻. Hydrolysis of [Xe₂F₁₁][WOF₅] by water contaminant in HF solvent yielded [XeF₅][WOF₅]⋅XeOF₄. Quantum-chemical calculations were carried out for M′OF₄, [M′OF₅]⁻, [M′₂O₂F₉]⁻, {[Xe₂F₁₁][CrOF₅]}₂, [Xe₂F₁₁][MOF₅], and {[XeF₅][M₂O₂F₉]}₂ to obtain their gas-phase geometries and vibrational frequencies to aid in their vibrational mode assignments and to assess chemical bonding.