On differentiability properties of typical continuous functions and Haar null sets

Let () be the set of all continuous functions on which have a derivative ( , respectively) at least at one point . B. R. Hunt (1994) proved that is Haar null (in Christensen's sense) in .

In the present article it is proved that neither nor its complement is Haar null in . Moreover, the same assertion holds if we consider the approximate derivative (or the ``strong' preponderant derivative) instead of the ordinary derivative; these results are proved using a new result on typical (in the sense of category) continuous functions, which is of interest in its own right.

     

The first member of the eleven-vertex azadicarbaborane series, 1,6,9-NC2B8H13, and its N-alkyl derivatives

Reactions between closo-1,2-C(2)B(8)H(10) (1) and amines of general formulation R(1)R(2)NH (where R(1), R(2) = H, H; Me, H; t-Bu, H and Et, Et) resulted in a straightforward cluster expansion and formation of the 11-vertex arachno-azadicarbaboranes of the 1,1-R(1),R(2-)1,6,9-NC(2)B(8)H(11) (2) cluster constitution (where R(1), R(2) = H, H 2a; Me, H 2b; t-Bu, H 2c and Et, Et 2d) in yields 10-75%, depending on the nature of the amine used. The reactions are the first example of a direct closo to arachno transformation in the area of cluster-boron compounds. Compounds 2b and 2c were isolated in two isomeric forms anti- and syn- that differ in the positioning of the t-Bu substituent with respect to the bridging hydrogen site. Deprotonation of compounds 2 generally leads to removal of the bridging proton and formation of the [1,1-R(1),R(2-)1,6,9-NC(2)B(8)H(11)](-) (2-) anions that, in the case of the monoalkylated Me and t-Bu derivatives, adopt only an anti configuration. The structure of anti-2c was determined by X-ray diffraction analysis and the geometries of the parent compound and the corresponding syn and anti isomers were optimised at the RMP2/6-31G* level. The composition of all compounds is consistent with the results of mass spectrometry and multinuclear ((1)H and (11)B) spectroscopy complemented by two-dimensional [(11)B-(11)B]-COSY and (1)H{(11)B(selective)} NMR measurements. Experimental (11)B chemical shifts generally show acceptable agreement with theoretical values calculated by GIAO methods, in particular at GIAO-MP2/II, where possible.     

Bifurcations and chaos in register transitions of excised larynx experiments

Experimental data from an excised larynx are analyzed in the light of nonlinear dynamics. The excised larynx provides an experimental framework that enables artificial control and direct observation of the vocal fold vibrations. Of particular interest in this experiment is the coexistence of two distinct vibration patterns, which closely resemble chest and falsetto registers of the human voice. Abrupt transitions between the two registers are typically accompanied by irregular vibrations. Two approaches are presented for the modeling of the excised larynx experiment; one is the nonlinear predictive modeling of the experimental time series and the other is the biomechanical modeling (three-mass model) that takes into account basic mechanisms of the vocal fold vibrations. The two approaches show that the chest and falsetto vibrations correspond to two coexisting limit cycles, which jump to each other with a change in the bifurcation parameter. Irregular vibrations observed at the register jumps are due to chaos that exists near the two limit cycles. This provides an alternative mechanism to generate chaotic vibrations in excised larynx experiment, which is different from the conventionally known mechanisms such as strong asymmetry between the left and right vocal folds or excessively high subglottal pressure.     

Water photodissociation in free ice nanoparticles at 243 nm and 193 nm

The photolysis of (H(2)O)(n) nanoparticles of various mean sizes between 85 and 670 has been studied in a molecular beam experiment. At the dissociation wavelength 243 nm (5.10 eV), a two-photon absorption leads to H-atom production. The measured kinetic energy distributions of H-fragments exhibit a peak of slow fragments below 0.4 eV with maximum at approximately 0.05 eV, and a tail of faster fragments extending to 1.5 eV. The dependence on the cluster size suggests that the former fragments originate from the photodissociation of an H(2)O molecule in the cluster interior leading to the H-fragment caging and eventually generation of a hydronium H(3)O molecule. The photolysis of surface molecules yields the faster fragments. At 193 nm (6.42 eV) a single photon process leads to a small signal from molecules directly photolyzed on the cluster surface. The two photon processes at this wavelength may lead to cluster ionization competing with its photodissociation, as suggested by the lack of H-fragment signal increase. The experimental findings are complemented by theoretical calculations.     

Remarkably efficient synthesis of 2H-indazole 1-oxides and 2H-indazoles via tandem carbon-carbon followed by nitrogen-nitrogen bond formation

Base-catalyzed tandem carbon-carbon followed by nitrogen-nitrogen bond formations quantitatively converted N-alkyl-2-nitro-N-(2-oxo-2-aryl-ethyl)-benzenesulfonamides to 2H-indazoles 1-oxides under mild conditions. Triphenylphosphine or mesyl chloride/triethylamine-mediated deoxygenation afforded 2H-indazoles.     

Emergence of charge-transfer-to-solvent band in the absorption spectra of hydrogen halides on ice nanoparticles: spectroscopic evidence for acidic dissociation

Extensive ab initio calculations complemented by a photodissociation experiment at 193 nm elucidate the nature of hydrogen halide molecules bound on free ice nanoparticles. Electronic absorption spectra of small water clusters (up to 5 water molecules) and water clusters doped with hydrogen fluoride, hydrogen chloride and hydrogen bromide were calculated. The spectra were modeled at the time-dependent density functional (TDDFT) level of theory with the BHandHLYP functional using the reflection principle. We observe the emergence of a charge-transfer-to-solvent (CTTS) band in the absorption spectra upon the acidic dissociation of the hydrogen halides. The CTTS band provides a spectroscopically observable feature for the acidic dissociation. The calculated spectra were compared with our new experimental photodissociation data for larger water clusters doped with HCl and HBr. We conclude that HCl and HBr dissociate to a large extent on the surface of ice nanoparticles at temperatures near 120 K and photoactive products are formed. The acidic dissociation of HX leads to an enhancement by about 4 orders of magnitude of the HCl photolysis rate in the 200-300 nm region, which is potentially relevant for the halogen budget in the atmosphere.     

High-performance liquid chromatographic/ion-trap mass spectrometric speciation of aquatic mercury as its pyrrolidinedithiocarbamate complexes

Mercury-pyrrolidinedithiocarbamate complexes are first time used for speciation of aquatic mercury with high-performance liquid chromatographic/ion trap-mass spectrometric method utilizing atmospheric pressure chemical ionization (APCI). The separation of the four mercury species was achieved in less than 5 min with a linear gradient profile of aqueous methanol from 70 up to 100% (v/v) in 4th min, isocratic elution at 100% up to 5th min and followed by a negative gradient to 70% in 6th min. The best separation was achieved on a reverse phase Zorbax Eclipse XDB C18 column (50 mm × 2.1 mm i.d., 1.8 μm particle size). The on-column limits of detection (injection volume 1 μL) were 370 pg for methylmercury (MeHg⁺), 280 pg for ethylmercury (EtHg⁺), 250 pg for phenylmercury (PhHg⁺) and 90 pg for inorganic mercury (Hg²⁺) when the data were collected in selective ion monitoring (SIM) mode. A method of isolation and preconcentration of the mercury species using a “home-made” C18 solid phase extraction (SPE) microcolumns was developed to enhance sensitivity of the method. The preconcentration factor as much as 2500 was achieved with on-column complex formation of mercury-pyrrolidinedithiocarbamate. Methanol (100%) was chosen for elution of preconcentrated mercury species. The method was applied for the determination of mercury species in river water samples.     

Experimental and theoretical study of the pyrrole cluster photochemistry: closing the pisigma* dissociation pathway by complexation

Photolysis of size selected pyrrole clusters has been investigated and compared to the photolysis of an isolated pyrrole molecule. Experimentally, size distributions of different mean cluster sizes (n=3 and n>5) have been prepared in supersonic expansions and the clusters were photolyzed at 243 and 193 nm. The kinetic energy distributions of the H photofragments have been measured. The distributions exhibit a bimodal character with fast and slow H-fragment peaks similar to the spectra of the bare molecule. However, with increasing cluster size the slow component gains intensity with respect to the fast one. A similar effect is observed with increasing the excitation energy from 243 to 193 nm. Theoretical calculations at the CASSCF/CASPT2 level have been performed for bare and complexed pyrroles (pyrrole is complexed with an argon atom and with another pyrrole unit). Combination of theoretical and experimental approaches leads to the conclusion that the direct dissociative pathway along the pisigma* potential energy surface in the N-H stretch coordinate is closed by the presence of the solvent molecule. This pathway is an important channel leading to the fast H atoms in the dissociation of the bare molecule. The solvent molecule influences significantly the electronic structure in the Rydberg-type pisigma* state while it has little influence on the valence states. The slow channel is mostly populated by the out-of-plane deformation mode which is also not influenced by solvation. We have also studied other possible reaction channels in pyrrole clusters (hydrogen transfer, dimerization). The present study shows that more insight into the bulk behavior of biologically relevant molecules can be gained from cluster studies.     

Combined isolation and purification procedures prior to the high-performance liquid chromatographic-ion-trap tandem mass spectrometric determination of estrogens and their conjugates in river sediments

A fast and sensitive analytical procedure has been developed for the simultaneous separation and determination of alpha-estradiol, beta-estradiol, estriol, estrone and ethynylestradiol and their sulfate, glucuronide and acetate conjugates in river sediments. The procedure includes a microwave-assisted extraction (MAE) with aqueous methanol (25:75, v/v) at 100 degrees C in 10 min, a clean-up on Oasis WAX cartridge and a high-performance liquid chromatography-ion-trap tandem mass spectrometry (HPLC-IT-MS/MS) with electrospray ionization. The recovery for each compounds ranged from 83 to 107% and the repeatability represented as RSDs ranged from 4.9 to 9.6%. The limits of detection (LODs) were down to 1 ng g(-1). The analytical performance of the method was evaluated using determination of free and conjugated estrogens in river sediments.     

Adsorption of poly(ethylene oxide)-block-polylactide copolymers on polylactide as studied by ATR-FTIR spectroscopy

In this study, the adsorption of amphiphilic poly(ethylene oxide)-block-polylactide (mPEO-PLA) copolymers from a selective solvent onto a polylactide surface was studied as a method of polylactide surface modification and its effect on nonspecific protein adsorption was evaluated. A series of well defined mPEO-PLA copolymers was prepared to investigate the effect of copolymer composition on the resulting PEO chain density and on the surface resistance to protein adsorption. The copolymers contained PEO blocks with molecular weights ranging between 5600 and 23,800 and with 16-47 wt% of PLA. The adsorption of both the copolymers and bovine serum albumin was quantified by attenuated total reflection FTIR spectroscopy (ATR-FTIR). In addition to the adsorbed copolymer amount, its actual composition was determined. The PEO chain density on the surface was found to decrease with the molecular weight of the PEO block and to increase with the molecular weight of the PLA block. The adsorbed copolymers displayed the ability to reduce protein adsorption. The maximum reduction within the tested series (by 80%) was achieved with the copolymer containing PEO of MW 5600 and a PLA block of the same MW.