On the Lipschitz classification of normed spaces, unit balls, and spheres

For every normed space , we note its closed unit ball and unit sphere by and , respectively. Let and be normed spaces such that is Lipschitz homeomorphic to , and is Lipschitz homeomorphic to .

We prove that the following are equivalent:

1. is Lipschitz homeomorphic to .

2. is Lipschitz homeomorphic to .

3. is Lipschitz homeomorphic to .

This result holds also in the uniform category, except (2 or 3) 1 which is known to be false.

     

A water-soluble and "self-assembled" polyoxometalate as a recyclable catalyst for oxidation of alcohols in water with hydrogen peroxide

We have demonstrated that a simply prepared water-soluble polyoxometalate, Na12[WZnZn2(H2O)2(ZnW9O34)2], synthesized from readily available zinc and tungsten salts in the presence of nitric acid, is an effective catalyst for selective alcohol oxidation with hydrogen peroxide in biphasic (water-alcohol) reaction media. Experiments have shown that the "self-assembled" catalyst in its mother liquor was as active as the isolated catalyst. The aqueous catalyst solution is easily separated from the water-insoluble products and can be recycled without loss in activity or selectivity.     

A family of 2D and 3D coordination polymers involving a trigonal tritopic linker

Five new coordination polymers, namely, [Zn(2)(H(2)O)(2)(BBC)](NO(3))(DEF)(6) (DUT-40), [Zn(3)(H(2)O)(3)(BBC)(2)] (DUT-41), [(C(2)H(5))(2)NH(2)][Zn(2)(BBC)(TDC)](DEF)(6)(H(2)O)(7) (DUT-42), [Zn(10)(BBC)(5)(BPDC)(2)(H(2)O)(10)](NO(3))(DEF)(28)(H(2)O)(8) (DUT-43), and [Co(2)(BBC)(NO(3))(DEF)(2)(H(2)O)](DEF)(6)(H(2)O) (DUT-44), where BBC--4,4',4'-(benzene-1,3,5-triyl-tris(benzene-4,1-diyl))tribenzoate, TDC--2,5-thiophenedicarboxylate, BPDC--4,4'-biphenyldicarboxylate, DEF--,N-diethylformamide, were obtained under solvothermal conditions and structurally characterized. It has been shown that compounds DUT-40, DUT-41 and DUT-44 exhibit 2D layered structures with large hexagonal channels. Utilization of additional angular dicarboxylic TDC linker led to the formation of the DUT-42 compound with the structure consisting of three interpenetrated 3D networks. Using the linear co-linker dicarboxylic BPDC, DUT-43 was obtained which forms a complicated 3D architecture arising from the polycatenation of triple-layered 2D building units and 2D single layer units. The pore accessibility of the synthesized compounds in the liquid phase was proved by the adsorption of dye molecules.     

Shotgun lipidomics on a LTQ Orbitrap mass spectrometer by successive switching between acquisition polarity modes

Top–down shotgun lipidomics relies on direct infusion of total lipid extracts into a high‐resolution tandem mass spectrometer and implies that individual lipids are recognized by their accurately determined m/z. Lipid ionization efficiency and detection specificity strongly depend on the acquisition polarity, and therefore it is beneficial to analyze lipid mixtures in both positive and negative modes. Hybrid LTQ Orbitrap mass spectrometers are widely applied in top–down lipidomics; however, rapid polarity switching was previously unfeasible because of the severe and immediate degradation of mass accuracy. Here, we report on a method to rapidly acquire high‐resolution spectra in both polarity modes with sub‐ppm mass accuracy and demonstrate that it not only simplifies and accelerates shotgun lipidomics analyses but also improves the lipidome coverage because more lipid classes and more individual species within each class are recognized. In this way, shotgun analysis of total lipid extracts of human blood plasma enabled to quantify 222 species from 15 major lipid classes within 7 min acquisition cycle. Copyright © 2012 John Wiley & Sons, Ltd.     

Chiral hydroperoxides as oxygen source in the catalytic stereoselective epoxidation of allylic alcohols by sandwich-type polyoxometalates: control of enantioselectivity through a metal-coordinated template

The epoxidation of allylic alcohols is shown to be efficiently and selectively catalyzed by the oxidatively resistant sandwich-type polyoxometalates, POMs, namely [WZnM(2)(ZnW(9)O(34))(2)](q)(-) [M = OV(IV), Mn(II), Ru(III), Fe(III), Pd(II), Pt(II), Zn(II); q = 10-12], with organic hydroperoxides as oxygen source. Conspicuous is the fact that the nature of the transition metal M in the central ring of polyoxometalate affects significantly the reactivity, chemoselectivity, regioselectivity, and stereoselectivity of the allylic alcohol epoxidation. For the first time, it is demonstrated that the oxovanadium(IV)-substituted POM, namely [ZnW(VO)(2)(ZnW(9)O(34))(2)](12-), is a highly chemoselective, regioselective, and also stereoselective catalyst for the clean epoxidation of allylic alcohols. A high enantioselectivity (er values up to 95:5) has been achieved with [ZnW(VO)(2)(ZnW(9)O(34))(2)](12)(-) and the sterically demanding TADOOL-derived hydroperoxide TADOOH as regenerative chiral oxygen source. Thus, a POM-catalyzed asymmetric epoxidation of excellent catalytic efficiency (up to 42 000 TON) has been made available for the development of sustainable oxidation processes. The high reactivity and selectivity of this unprecedented oxygen-transfer process are mechanistically rationalized in terms of a peroxy-type vanadium(V) template.     

Analysis of cocaine and its principal metabolites in waste and surface water using solid-phase extraction and liquid chromatography–ion trap tandem mass spectrometry

A validated method based on solid-phase extraction (SPE) and liquid chromatography-ion trap tandem mass spectrometry (LC-MS/MS) is described for the determination of cocaine (COC) and its principal metabolites, benzoylecgonine (BE) and ecgonine methyl ester (EME), in waste and surface water. Several SPE adsorbents were investigated and the highest recoveries (95.7 +/- 5.5, 91.8 +/- 2.2 and 72.5 +/- 5.3% for COC, BE and EME, respectively) were obtained for OASIS HLB(R) cartridges (6 mL/500 mg) using 100 mL of waste water or 500 mL of surface water. Extracts were analysed by reversed-phase (RP) or hydrophilic interaction (HILIC) LC-MS/MS in positive ion mode with multiple reactions monitoring (MRM); the latter is the first reported application of the HILIC technique for drugs of abuse in water samples. Corresponding deuterated internal standards were used for quantification. The method limits of quantification (LOQs) for COC and BE were 4 and 2 ng L(-1), respectively, when RPLC was used and 1, 0.5 and 20 ng L(-1) for COC, BE and EME, respectively, with the HILIC setup. For COC and BE, the LOQs were below the concentrations measured in real water samples. Stability tests were conducted to establish the optimal conditions for sample storage (pH, temperature and time). The degradation of COC was minimal at -20 degrees C and pH = 2, but it was substantial at +20 degrees C and pH = 6. The validated method was applied to a set of waste and surface water samples collected in Belgium.     

Interaction of scanning probes with semiconductor nanocrystals; physical mechanism and basis for near-field optical imaging

We investigate the modification of photoluminescence (PL) from single semiconductor nanocrystal quantum dots (NCs) in the proximity of metal and semiconducting atomic force microscope (AFM) tips. The presence of the tip alters the radiative decay rate of an emitter via interference and opens efficient nonradiative decay channels via energy transfer to the tip material. These effects cause quenching (or enhancement) of the emitter's PL intensity as a function of its distance from the interacting tip. We take advantage of this highly distance-dependent effect to realize a contrast mechanism for high-resolution optical imaging. AFM tips are optimized as energy acceptors by chemical functionalization with InAs NCs to achieve optical resolution down to 30 nm. The presented experimental scheme offers high-resolution optical information while maintaining the benefits of traditional AFM imaging. We directly measure the PL intensity of single NCs as a function of the tip distance. Our results are in good agreement with calculations made by a classical theoretical model describing an oscillating dipole interacting with a planar mirror.