Infinitely many leaf-wise intersections on cotangent bundles

If the homology of the free loop space of a closed manifold B$B$ is infinite dimensional then generically there exist infinitely many leaf-wise intersection points for fiberwise star-shaped hypersurfaces in T^∗B$T^{\ast }B$. We illustrate this in the case of the restricted three body problem.     

Die eindeutige Primfaktorzerlegung

Zusammenfassung  Die Frage der Primfaktorzerlegung in Unterringen der komplexen Zahlen und der unmittelbar damit zusammenh?ngenden S?tze wird in der heutigen Algebra ohne grossen Aufwand und fast nebenbei behandelt: Studierende haben damit auch kaum Schwierigkeiten. In der Geschichte allerdings verlief die Entwicklung alles andere als gradlinig. Ein genauerer Blick auf die historischen Einzelheiten erlaubt interessante und in vielerlei Hinsicht überraschende Einsichten in die vertrackte Art und Weise, wie sich Mathematik manchmal entwickelt. Hier soll diese Geschichte erz?hlt werden, wie sie sich aus den neueren mathematikhistorischen Forschungen von H.M. Edwards, R. B?lling, O. Neumann und F. Lemmermeyer ergibt, und zwar auf einem Niveau, das einem Mathematikstudierenden nach einer Algebra-Vorlesung zug?nglich ist.     

One-Dimensional Hydrogen-Bonded Structures in the 1:1 Proton-Transfer Salts of 4,5-Dichlorophthalic Acid with the Aliphatic Lewis Bases Diisopropylamine and Hexamethylenetetramine

Abstract  

The crystal structures of the 1:1 proton-transfer compounds of 4,5-dichlorophthalic acid with the aliphatic Lewis bases diisopropylamine and hexamethylenetetramine, viz. diisopropylaminium 2-carboxy-4,5-dichlorobenzoate (1) and hexamethylenetetraminium 2-carboxy-4,5-dichlorobenzoate hemihydrate (2), have been determined. Crystals of both 1 and 2 are triclinic, space group P − 1, with Z = 2 in cells with a = 7.0299(5), b = 9.4712(7), c = 12.790(1) ?, α = 99.476(6), β = 100.843(6), γ = 97.578(6)° (1) and a = 7.5624(8), b = 9.8918(8), c = 11.5881(16) ?, α = 65.660(6), β = 86.583(4), γ = 86.987(8)° (2). In each, one-dimensional hydrogen-bonded chain structures are found: in 1 formed through aminium N⁺–H…O_carboxyl cation–anion interactions. In 2, the chains are formed through anion carboxyl O···H–O_{bridging water} interactions with the cations peripherally bound. In both structures, the hydrogen phthalate anions are essentially planar with short intra-species carboxylic acid O–H…O_carboxyl hydrogen bonds [O…O, 2.381(3) ? (1) and 2.381(8) ? (2)].     

咪唑基离子液体的物理化学性质估算及预测(英文)

根据经验和半经验方程及空隙模型理论,可以估算及预测离子液体在298.15K的物理化学性质.本文讨论了离子液体的分子体积,密度,标准熵,晶格能,表面张力,等张比容,摩尔蒸发焓,空隙体积,空隙率和热膨胀系数.通过实验测得的三种离子液体1-乙基-3-甲基咪唑硫酸乙酯([C2mim][EtSO4)]),1-丁基-3-甲基咪唑硫酸辛酯([C4mim][OcSO4])和1-乙基-3-甲基咪唑双三氟甲磺酰亚胺盐([C2mim][NTf2])的密度和表面张力估算了它们的其它物理化学性质.由这三种离子液体的分子体积及等张比容预测了同系列中其它离子液体[Cnmim][EtSO4],[Cnmim][OcSO4]和[Cnmim][NTf2](n=1-6)的分子体积及等张比容,由此计算出它们的密度及表面张力.进而预测了它们的物理化学性质.将预测的离子液体[C4mim][NTf2]和[C2mim][OcSO4]的密度值与文献报导的实验值进行比较,其偏差在实验误差范围内.最后,将由Kabo经验方程计算的七个离子液体[C2mim][EtSO4]、[C4mim][OcSO4]、[C2mim][NTf2]、[C4mim][NTf2]、丁基三甲基铵双三氟甲磺酰亚胺盐([N4111][NTf2])、甲基三辛基铵双三氟甲磺酰亚胺盐([N8881][NTf2])和1-辛基-3-甲基吡啶四氟硼酸盐([m3opy][BF4])的摩尔蒸发焓与由Verevkin简单规则预测的摩尔蒸发焓进行比较,发现两者符合很好.因此,在缺乏密度和表面张力实验数据的情况下,可以用Verevkin简单规则来预测离子液体的摩尔蒸发焓.     

Spectroscopical and Electrochemical Behavior of New Mixed-Ligand Cyclometalated Rh(III) Complexes

The absorption spectra, luminescence spectra, excited-state lifetimes, and electrochemical behavior of the cyclometalated [Rh(ppz)₂bpy]+^, [Rh(3-Cl-ppz)₂(bpy)]⁺, [Rh(4-NO2-ppz)₂(bpy)]⁺, [Rh(ppz)₂(biq)]⁺ and [Rh(4-NO2-ppz)₂(biq)]⁺ complexes (ppz^?, 3-Cl-ppz^?, and 4-NO₂-ppz^? are the ortho-C-deprotonated forms of 1-phenylpyrazole, l-(3-chlorophenyl)pyrazole and l-(4-nitrophenyl)pyrazole, respectively) have been investigated. The results obtained have been compared with those concerning the free protonated ligands and some previously studied mixed-ligand cyclometalated Rh(III) complexes. Luminescence originates from the lowest ligand-centered (LC) excited state, which involves the diimine ligands in all cases except for [Rh(4-NO2-ppz)₂(bpy)]⁺, where it involves the ortho-metalating ligand. s. In the absorption spectra, LC and metal-to-ligand charge-transfer (MLCT) bands, involving the diimine and/or the ortho-metalating ligands, have been assigned, and correlations between spectroscopic and electrochemical data are discussed.     

Complexes of Rhodium(III) with Two Chelating C⁁N Ligands and One Diimine Ligand

The synthesis of two series of cyclometalated Rh^III compounds is described, namely of 5 dinuclear chlorobridged species [{Rh(C?N)₂} (μ-Cl)₂] A and of 15 mononuclear complexes [Rh(C?N)₂(N?N)]⁺ B ; C?N stands for five different cyclometalating ligands, i.e. deprotonated phenylpyridine, deprotonated 2-(thienyl)pyridine, and three deprotonated 1-aryl-1H-pyrazoles, and N?N for six diimine ligands such as 2,2-bihyridine, 2,2′-bi-1H-imidazole, and 2,2′-bipyrimidine. For (2,2′-bipyridine)bis[2-(thien-2-yl)pyridinato-N,C^3′]rhodium(III) chloride, an X-ray structure determination was carried out. In the other cases, ¹H-NMR spectra established the configuration of the complexes. All mononuclear and dinuclear complexes show a C,C cis-configuration. The UV/VIS-absorption bands at longest wavelength are most likely due to metal-to-ligand charge-transfer (MLCT) transitions, depending on the nature of the cyclometalating ligand C?N and on the diimine ligand N?N. The receptor orbital is in some cases on the cyclometalating ligand, in others on the diimine. All monomer complexes exhibit at least one reversible reduction wave in the cyclovoltammogram in dimethylformamide solutions, attributable to a ligand-centered reduction. It is, therefore, concluded that the LUMO in [Rh(C?N)₂(N?N)]⁺ is of L(π*) character, as opposed to [Rh(bpy)₃]³⁺, where it is a metal d-orbital. The crystal system of (2,2′-bipyridine)bis[2-(thien-2-yl)pyridinato-N,C_3′]rhodium(III) chloride—water (1/2.125) is tetragonal (space group P⁴; R = 0.036, R_w = 0.040). The Rh-atom has slightly distorted octahedral environment; the average distances are (Rh? N/thienyl? pyridine) = 2.060 (3), Rh? C = 1.9885 (3), and Rh? N(bipyridine) = 2.1415 (3) Å. Of the three ligands the 2,2′-bipyridine is the most planar.     

Dirichlet problem at infinity on Gromov hyperbolic metric measure spaces

We study the asymptotic Dirichlet problem for p-harmonic functions in a very general setting of Gromov hyperbolic metric measure spaces.     

An improved method for the analysis of volatile polyfluorinated alkyl substances in environmental air samples

This article describes the optimisation and validation of an analytical method for the determination of volatile polyfluorinated alkyl substances (PFAS) in environmental air samples. Airborne fluorinated telomer alcohols (FTOHs) as well as fluorinated sulfonamides and sulfonamidoethanols (FOSAs/FOSEs) were enriched on glass-fibre filters (GFFs), polyurethane foams (PUFs) and XAD-2 resin by means of high-volume air samplers. Sensitive and selective determination was performed using gas chromatography/chemical ionisation–mass spectrometry (GC/CI–MS). Five mass-labelled internal standard (IS) compounds were applied to ensure the accuracy of the analytical results. No major blank problems were encountered. Recovery experiments were performed, showing losses of the most volatile compounds during extraction and extract concentration as well as strong signal enhancement for FOSEs due to matrix effects. Breakthrough experiments revealed losses of the most volatile FTOHs during sampling, while FOSAs/FOSEs were quantitatively retained. Both analyte losses and matrix effects could be remediated by application of adequate mass-labelled IS. Method quantification limits (MQLs) of the optimised method ranged from 0.2 to 2.5 pg/m³ for individual target compounds. As part of the method validation, an interlaboratory comparison of instrumental quantification methods was conducted. The applicability of the method was demonstrated by means of environmental air samples from an urban and a rural location in Northern Germany. Figure High-volume air sampling of volatile polyfluorinated alkyl substances using glass fibre filters and PUF/XAD-2 cartridges at a background monitoring site (Waldhof, Germany)