Synthesis and characterization of poly(ethylene glycol) based β-cyclodextrin polymers

A series of novel water soluble β-cyclodextrin (βCD) polymers has been synthesized from functionalized poly(ethylene glycol) (PEG). The chemical composition of the polymers has been characterized by ¹H NMR and the βCD content is found to be between 48 and 33% (w/w). The molecular weight has been determined by Size Exclusion Chromatography (SEC) and depends on the ratio between βCD and PEG, varying from 2.1 × 10⁴ to 8.6 × 10⁴ g mol^?1. The physico chemical properties have been characterized by differential scanning calorimetry (DSC), viscometry and isothermal titration calorimetry (ITC). ITC shows that the polymers have association constants comparable to βCD with different guest molecules, indicating a good accessibility of the CDs.     

Liquid-phase microextraction in a microfluidic-chip – High enrichment and sample clean-up from small sample volumes based on three-phase extraction

In this work, a microfluidic-chip based system for liquid-phase microextraction (LPME-chip) was developed. Sample solutions were pumped into the LPME-chip with a micro-syringe pump at a flow rate of 3–4 μL min⁻¹. Inside the LPME chip, the sample was in direct contact with a supported liquid membrane (SLM) composed of 0.2 μL dodecyl acetate immobilized in the pores of a flat membrane of polypropylene (25 μm thickness). On the other side of the SLM, the acceptor phase was present. The acceptor phase was either pumped at 1 μL min⁻¹ during extraction or kept stagnant (stop-flow). Amitriptyline, methadone, haloperidol, loperamide, and pethidine were selected as model analytes, and they were extracted from alkaline sample solution, through the SLM, and into 10 mM HCl or 100 mM HCOOH functioning as acceptor phase. Subsequently, the acceptor phase was either analyzed off-line by capillary electrophoresis for exact quantification, or on-line by UV detection or electrospray ionization mass spectrometry for time profiling of concentrations. The LPME-chip was found to be highly effective, and extraction efficiencies were in the range of 52–91%. When the flow of acceptor phase was turned off during extraction (stop-flow), analyte enrichment increased linearly with the extraction time. After 10 min as an example, amitriptyline was enriched by a factor of 42 from only 30 μL sample solution, and after 120 min amitriptyline was enriched by a factor of 500 from 320 μL sample solution. This suggested that the LPME-chip has great potentials for very efficient analyte enrichments from limited sample volumes in the future.     

Stabilisation of [Fe2(CO)9] and [Ru2(CO)9] bu substitution with bridging diphosphorus ligands

Reaction of [Fe₂(CO)₉] with a half molar amount of R₂PYPR₂ (Y = CH₂, R = Ph, Me, OMe or OPrⁱ; Y = N(Et), R = OPh, OMe or OCH₂; Y = N(Me), R = OPrⁱ or OEt) leads to the ready formation of a product which on irradiation with ultraviolet light rapidly decarbonylates to the heptacarbonyl derivative [Fe₂(μ-CO)(CO)₆{μ-R₂PYPR₂}]. Treatment of the latter with a slight excess of the appropriate ligand results, under photochemical conditions, in the formation of the dinuclear pentacarbonyl complex [Fe₂(μ-CO)(C))₄{μ-R₂PYPR₂}₂] but under thermal conditions in the formation of the mononuclear species [Fe(CO)₃{R₂PYPR₂}]. Reaction of [Ru₃(CO)₁₂] with an equimolar amount of (RO)₂PN(R′)P(OR)₂ (R′ = Me, R = Prⁱ or Et; R′ = Et, R = Ph or Me) under either thermal or photochemical conditions produces [Ru₃(CO)₁₀{μ-(RO)₂PN(OR)₂}] which reacts further with excess (RO)₂PN(R′)P(OR)₂ on irradiation with ultraviolet light to afford the dinuclear compound [Ru₂(μ-CO)(CO₄{μ-(RO)₂PN(R′)P(OR)₂}₂]. The molecular structure of [Ru₂(μ-CO)(CO)₄{μ-(MeO)₂PN(Et)P(OMe)₂}₂], which has been determined by X-ray crystallography, is described.     

The dual spectral set conjecture

Suppose that where are real numbers such that and The union is not assumed to be disjoint. It is shown that the translates , , tile the real line for some bounded measurable set if and only if the exponentials , , form an orthogonal basis for some bounded measurable set

     

Minimal webs in Riemannian manifolds

For a given combinatorial graph G a geometrization (G, g) of the graph is obtained by considering each edge of the graph as a 1-dimensional manifold with an associated metric g. In this paper we are concerned with minimal isometric immersions of geometrized graphs (G, g) into Riemannian manifolds (N ⁿ , h). Such immersions we call minimal webs. They admit a natural ‘geometric’ extension of the intrinsic combinatorial discrete Laplacian. The geometric Laplacian on minimal webs enjoys standard properties such as the maximum principle and the divergence theorems, which are of instrumental importance for the applications. We apply these properties to show that minimal webs in ambient Riemannian spaces share several analytic and geometric properties with their smooth (minimal submanifold) counterparts in such spaces. In particular we use appropriate versions of the divergence theorems together with the comparison techniques for distance functions in Riemannian geometry and obtain bounds for the first Dirichlet eigenvalues, the exit times and the capacities as well as isoperimetric type inequalities for so-called extrinsic R-webs of minimal webs in ambient Riemannian manifolds with bounded curvature.      

Graded cellular bases for Temperley–Lieb algebras of type A and B

We show that the Temperley–Lieb algebra of type A and the blob algebra (also known as the Temperley–Lieb algebra of type B) at roots of unity are \(\mathbb{Z}\) -graded algebras. We moreover show that they are graded cellular algebras, thus making their cell modules, or standard modules, graded modules for the algebras.     

Finite metric spaces of strictly negative type

We prove that, if a finite metric space is of strictly negative type, then its transfinite diameter is uniquely realized by the infinite extender (load vector). Finite metric spaces that have this property include all spaces on two, three, or four points, all trees, and all finite subspaces of Euclidean spaces. We prove that, if the distance matrix is both hypermetric and regular, then it is of strictly negative type. We show that the strictly negative type finite subspaces of spheres are precisely those which do not contain two pairs of antipodal points. In connection with an open problem raised by Kelly, we conjecture that all finite subspaces of hyperbolic spaces are hypermetric and regular, and hence of strictly negative type.     

Characterizations by Automorphism Groups of some Rank 3 Buildings – I. Some Properties of Half Strongly-Transitive Triangle Buildings

In a sequence of papers, we will show that the existence of a (half) strongly-transitive automorphism group acting on a locally finite triangle building forces to be one of the examples arising from PSL₃(K) for a locally finite local skewfield K. Furthermore, we introduce some Moufang-like conditions in affine buildings of rank 3, and characterize those examples arising from algebraic, classical or mixed type groups over a local field. In particular, we characterize the p-adic-like affine rank 3 buildings by a certain p-adic Moufang condition, and show that such a condition has zero probability to survive in hyperbolic rank 3 buildings. This shows that a construction of hyperbolic buildings as analogues of p-adic affine buildings is very unlikely to exist.