Frames and the Feichtinger conjecture

We show that the conjectured generalization of the Bourgain-Tzafriri restricted-invertibility theorem is equivalent to the conjecture of Feichtinger, stating that every bounded frame can be written as a finite union of Riesz basic sequences. We prove that any bounded frame can at least be written as a finite union of linearly independent sequences. We further show that the two conjectures are implied by the paving conjecture. Finally, we show that Weyl-Heisenberg frames over rational lattices are finite unions of Riesz basic sequences.

     

Asymptotic behavior of tails and quantiles of quadratic forms of Gaussian vectors

We derive results on the asymptotic behavior of tails and quantiles of quadratic forms of Gaussian vectors. They appear in particular in delta–gamma models in financial risk management approximating portfolio returns. Quantile estimation corresponds to the estimation of the Value-at-Risk, which is a serious problem in high dimension.     

Chaotic resonance: A simulation

Stochastic resonance is a statistical phenomenon that has been observed in periodically modulated, noise-driven, bistable systems. The characteristic signatures of the effect include an increase in the signal-to-noise of the output as noise is added to the system, and exponentially decreasing peaks in the probability density as a function of residence times in one state. Presented are the results of a numerical simulation where these same signatures were observed by adding achaotic driving term instead of a white noise term. Although the probability distributions of the noise and chaos inputs were significantly different, the stochastic and chaotic resonances were equal within the experimental error.     

Quantification of midodrine and its active metabolite in plasma using a high performance liquid chromatography column switching technique

An automated column-switching HPLC system is described for the simultaneous determination of midodrine, an alpha-adrenergic stimulating drug, and its active metabolite, ST-1059. Serum or plasma (850 microliters) is directly injected onto a RP18 (30 micrograms particle size) pre-column (9 x 4 mm ID) which acts as an on-line liquid-solid extractor and analyte enrichment system. The injection is followed by washing steps. The fraction containing the analytes is transferred onto an analytical RP18 column via step gradient elution where the final analysis is performed. Fluorescence detection is used (lambda ex 290 nm and lambda em 322 nm), and method detection limits of 0.8 ng/mL plasma were reached. These were sufficiently low to determine the plasma concentration-time profiles for both compounds following oral administration of 2.5 mg and 5 mg midodrine hydrochloride. The assay in serum or plasma was linear in the range of 1 to 15 ng analyte/mL, the recovery was greater than 95%, and the reproducibility was sufficient. The assay was rugged and was maintained by routinely changing the home-made, dry packed pre-column every 20th serum injection.     

Development of reactive thiol-modified monolithic capillaries and in-column surface functionalization by radical addition of a chromatographic ligand for capillary electrochromatography

Reactive thiol-modified capillary columns for capillary electrochromatography (CEC) were developed by transforming the pendent 2,3-epoxypropyl groups of poly(glycidyl methacrylate-co-ethylene dimethacrylate) (poly(GMA-co-EDMA)) monoliths into 3-mercapto-2-hydroxy-propyl residues by a nucleophilic substitution reaction, employing sodium-hydrogen sulfide as nucleophilic reagent. Conditions for this modification reaction were systematically optimized with respect to different parameters, such as reaction temperature, pH-value, reaction time, type and concentration of organic modifier, and concentration of the sodium-hydrogen sulfide solution. The amount of thiol groups that was generated on the monolith surface was determined directly in the capillaries by a disulfide-exchange reaction employing 2,2'-dipyridyl disulfide (DPDS). This reaction in the capillary liberates pyridine-2-thione in equimolar amount to the surface sulfhydryls, which was collected into a vial and determined photometrically at 343 nm by RP-HPLC. About 17% of the total lateral epoxide moieties of the monolithic substrate could be transformed to reactive sulfhydryl groups, which corresponds to about 0.7 mmol g(-1) monolithic polymer, with a column-to-column repeatability of 3.2% R.S.D. The reactive thiol groups can be utilized to attach any chromatographic ligand with appropriate anchor in a second step, e.g. by radical addition, graft polymerization, nucleophilic substitution, disulfide formation or Michael addition reaction. To demonstrate the feasibility of the concept, we chose an anion exchange type chromatographic ligand based on a quinine derivative, O-9-tert-butylcarbamoylquinine (t-BuCQN) which was attached to the monolith in a radical addition reaction, for a further in-column surface functionalisation. About 78% of the sulfhydryl groups were derivatized with t-BuCQN as determined from differential DPDS assays before and after the selector immobilization reaction. The applicability of these surface-functionalised monolithic capillary columns could be shown by an electrochromatographic separation of the enantiomers of N-3,5-dinitrobenzoyl-leucine, which performed fairly well compared to an analogous capillary that was fabricated by an in situ copolymerization approach.