Harmonic analysis of fractal measures

We consider affine systems inR ⁿ constructed from a given integral invertible and expansive matrixR, and a finite setB of translates,σ _bx:=R^–1x+b; the corresponding measure μ onR ⁿ is a probability measure fixed by the self-similarity $\mu = \left| B \right|^{ - 1} \sum\nolimits_{b \in B} {\mu o\sigma _b^{ - 1} } $ . There are twoa priori candidates for an associated orthogonal harmonic analysis: (i) the existence of some subset Λ inR ⁿ such that the exponentials {e^iλ·x}Λ form anorthogonal basis forL ²(μ); and (ii) the existence of a certaindual pair of representations of theC ^*-algebraO _N wheren is the cardinality of the setB. (For eachN, theC ^*-algebraO _N is known to be simple; it is also called the Cuntz algebra.) We show that, in the “typical” fractal case, the naive version (i) must be rejected; typically the orthogonal exponentials inL ²(μ) fail to span a dense subspace. Instead we show that theC ^*-algebraic version of an orthogonal harmonic analysis, namely (ii), is a natural substitute. It turns out that this version is still based on exponentialse ^iλ·x, but in a more indirect way. (See details in Section 5 below.) Our main result concerns the intrinsic geometric features of affine systems, based onR andB, such that μ has theC ^*-algebra property (ii). Specifically, we show that μ has an orthogonal harmonic analysis (in the sense (ii)) if the system (R, B) satisfies some specific symmetry conditions (which are geometric in nature). Our conditions for (ii) are stated in terms of two pieces of data: (a) aunitary generalized Hadamard matrix, and (b) a certainsystem of lattices which must exist and, at the same time, be compatible with the Hadamard matrix. A partial converse to this result is also given. Several examples are calculated, and a new maximality condition for exponentials is identified.     

Liquid chromatography electrospray mass spectrometry with variable fragmentor voltages gives simultaneous elemental and molecular detection of arsenic compounds

A single quadrupole high performance liquid chromatography electrospray mass spectrometry system with a variable fragmentor voltage facility was used in the positive ion mode for simultaneous recording of elemental and molecular mass spectral data for arsenic compounds. The method was applicable to the seven organoarsenic compounds tested: four arsenic-containing carbohydrates (arsenosugars), a quaternary arsonium compound (arsenobetaine), dimethylarsinic acid, and dimethylarsinoylacetic acid. It was not suitable for the two inorganic arsenic species arsenite and arsenate. In the case of arsenosugars, qualifying ion data for a characteristic common fragment (m/z 237) was also simultaneously obtained. The method was used to identify and quantify the major arsenosugars in crude extracts of two brown algae.     

Procedure for background estimation in energy-dispersive X-ray fluorescence spectra

A method for semi-automatic background estimation in energy-dispersive x-ray fluorescence spectra is outlined. Two cubic splines were investigated and the spline called the Butland interpolant was chosen for further investigation. Prior to the calculation of the spline, peak regions are set up, and suitable knots are defined outside the peak regions. To set up the peak regions, an automatic peak-search routine and a calibration equation are used. For a given peak, the latter relates the full width at half maximum (FWHM) to the peak centre. In turn, the size of the peak regions are defined by the FWHM multiplied by a constant given by the user. The method was tested on several types of spectra. It was found that the optimal size of the peak region decreased with increasing peak density. Reproducibility tests showed that the standard deviation of the summation of counts within a peak region and after background subtraction was less than would be expected from the counting statistics.     

Apparent Specific Volume Measurements of Poly(ethylene oxide), Poly(butylene oxide), Poly(propylene oxide), and Octadecyl Chains in the Micellar State as a Function of Temperature

Apparent specific densities of aqueous solutions of the diblock copolymers C18(EO)100, C18(EO)20, and (EO)92(BO)18 and the triblock copolymers (EO)25(PO)40(EO)25 and (EO)21(PO)47(EO)21 in the micellar state have been measured over a temperature range from 10 to 90 degrees C at concentrations between 1% and 5%, using an oscillating tube densitometer. From these measurements, apparent specific volumes of poly(ethylene oxide) (PEO), poly(propylene oxide) (PPO), poly(butylene oxide) (PBO), and octadecane in the micellar state have been determined. The composition of the block copolymers was checked by NMR spectroscopy. Results were compared with published data for the polymers and bulk values for octadecane, respectively. The apparent specific density of PEO chains in the dissolved state was also measured for PEG4600 solutions at different concentrations and compared with results in the micellar state. The results presented in the paper are crucial in connection with analysis and modeling of small-angle X-ray scattering (SAXS) data from polymer and block copolymer micellar systems. PEO and PPO have a relatively low apparent partial specific volume in water at low temperatures. It is associated with water molecules making strong hydrogen bonds with the oxygen atoms on the polymer backbone. These water molecules gradually become disordered when the temperature is increased and the polymer apparent specific volume increases. For PBO in the micellar cores of PBO-PEO block copolymer micelles and in PNiPAM microgels, pronounced temperature dependence with the same origin is also found. The application of the derived results for the apparent specific volume of PEO for deriving contrast factors is demonstrated and the results are used in the analysis of SAXS data for semidilute solutions of PEG4600 in a broad temperature range.     

Preparation temperature dependence of size and polydispersity of alkylthiol monolayer protected gold clusters

The influence of preparation temperature on the size and size distribution of dodecylthiol monolayer protected gold clusters was studied. The monolayer protected clusters (MPCs) were synthesized by two different variations of the Brust-Schiffrin procedure. In all of the experiments, the stoichiometry of the reactants dodecylthiol, HAuCl(4), and sodium borohydride was kept constant, while the temperature was varied in the range of -18 to +90 degrees C. Two series were performed in which an aqueous solution of NaBH(4) was either added over 30 s or all in one portion. The size and size distribution of the MPCs were determined by small-angle X-ray scattering (SAXS) and transmission electron microscopy (TEM). It has been demonstrated that in general the MPC size increases with elevated preparation temperatures.     

Combinatorial synthesis of benztropine libraries and their evaluation as monoamine transporter inhibitors

A combinatorial synthesis of benztropine analogues is presented. Radical azidonation of 3-benzyloxy-8-azabicyclo[3.2.1]octane-8-carboxylic acid tert-butyl ester 3 to 3-(1-azidobenzyloxy)-8-azabicyclo[3.2.1]octane-8-carboxylic acid tert-butyl ester 4 was used as a key step in the synthesis. This step was optimized by adding 10% DMF to the reaction. Reaction of 4 with phenyl magnesium bromide followed by Boc removal and N-methylation gave benztropine 1. Reaction of five-component Grignard reagents with 4 was used to create a two-dimensional library of 25 N-normethylbenztropine analogues. Further reaction of this library with five alkyl bromides was carried out to create a three-dimensional library containing 125 compounds. Screening of the libraries towards binding and inhibition of uptake of the human dopamine (hDAT), serotonin (hSERT) and norepinephrine transporters (hNET) was carried out. None of the synthesized compounds were found to be stronger than benztropine, and none were selective for inhibition of binding over monoamine uptake.     

Synthesis of Acyclic Nitroazole Nucleosides and Their Incorporation into Oligonucleotides,and Their Duplex and Triplex Formation

Acyclic nucleosides of 4‐nitro‐1H‐imidazole and 4‐nitropyrazole have been synthesized by nucleophilic addition of the appropriate 4‐nitroazole to (?)‐(S)‐(hydroxymethyl)oxirane in the presence of a catalytic amount of potassium carbonate. (+)‐(R)‐3‐(4‐nitro‐1H‐imidazol‐1‐yl)propane‐1,2‐diol and (+)‐(R)‐3‐(2‐methyl‐4‐nitro‐1H‐imidazol‐1‐yl)propane‐1,2‐diol were also obtained in an independent reaction starting from appropriate 1,4‐dinitro‐1H‐imidazole and (+)‐(R)‐3‐aminopropane‐1,2‐diol. (+)‐(R)‐3‐(4‐Nitropyrazol‐1‐yl)propane‐1,2‐diol was also obtained by direct noncatalyzed addition of 4‐nitropyrazole to (?)‐(S)‐(hydroxymethyl)oxirane, whereas the (S)‐enantiomer was obtained by reaction of 4‐nitropyrazole with (+)‐(S)‐1,2‐O‐isopropylideneglycerol under Mitsunobu reaction conditions, followed by a cleavage of the isopropylidene group with 80% AcOH. Racemization during any of these syntheses has not been observed. 3‐(4‐Nitroazol‐1‐yl)propane‐1,2‐diols were incorporated into a 26‐mer oligonucleotide. UV Thermal melting studies of duplexes of the oligonucleotides with 4‐nitropyrazole or 4‐nitro‐1H‐imidazole paired with four natural bases showed moderately decreased stabilities of the duplexes. A narrow range of melting temperatures, typically being within 2° for each acyclic nucleoside, fulfill one of the requirements of using acyclic 4‐nitroazoles as general bases. Single incorporation of 4‐nitroazoles into a 14‐mer triplex forming oligonucleotide resulted in considerably decreased triplex stabilities.     

HPLC Method for Determination of Rosiglitazone in Plasma

A fast and sensitive high performance liquid chromatography method for quantitative determination of rosiglitazone in human plasma has been developed. The extraction from plasma was performed using solid-phase extraction (SPE) on C4 silica (100 mg) disposable extraction cartridges (DEC). The separation of rosiglitazone and two metabolites was achieved on a Phenomenex® Synergi 4 µm MAX-RP (150 × 4.6 mm) column, protected by a guard column. The mobile phase was 0.01 M ammonium acetate, pH 7.0 - acetonitrile (65:35, v/v). (3S)-3-OH-quinidine was used as internal standard. The analytes were detected using fluorescence detection. The method was validated. The limit of quantitation was 1 ng mL⁻¹ and the detection limit was 0.25 ng mL⁻¹ for rosiglitazone in human plasma. The recovery was 90% for rosiglitazone. Linearity was observed over a range of 1-1000 ng mL⁻¹ (r²=0.9959). The intra- and inter-day precision (C.V.) did not exceed 8.7 %. Applicability of the method was demonstrated by a clinical pharmacokinetic study. A healthy volunteer received in two separate phases 4 mg and 8 mg rosiglitazone maleate as a single oral dose. Plasma concentrations were measured for 24 h in both phases.