Geometric Structures on the Complement of a Projective Arrangement

Consider a complex projective space with its Fubini-Study metric. We study certain one parameter deformations of this metric on the complement of an arrangement (= finite union of hyperplanes) whose Levi-Civita connection is of Dunkl type. Interesting examples are obtained from the arrangements defined by finite complex reflection groups. We determine a parameter interval for which the metric is locally of Fubini-Study type, flat, or complex-hyperbolic. We find a finite subset of this interval for which we get a complete orbifold or at least a Zariski open subset thereof, and we analyze these cases in some detail (e.g., we determine their orbifold fundamental group).In this set-up, the principal results of Deligne-Mostow on the Lauricella hypergeometric differential equation and work of Barthel-Hirzebruch-Höfer on arrangements in a projective plane appear as special cases. Along the way we produce in a geometric manner all the pairs of complex reflection groups with isomorphic discriminants, thus providing a uniform approach to work of Orlik-Solomon.     

Estimation of surface desorption times in hydrophobically coated nanochannels and their effect on shear-driven and pressure-driven chromatography

The present paper provides a detailed analysis of the analyte-wall adsorption effects in nanochannels, including a random walk study of the analyte-wall collision frequency, and uses these insights to estimate wall desorption times from chromatographic experiments in nanochannels. Using coumarin dye analytes and using a methanol/water mixture buffered at pH 3 in 120-nm deep channels, the surface desorption times on naked fused-silica glass were found to be maximally of the order of 60 to 150 μs, while they were found to be on the order of 100 to 500 μs on a hydrophobically coated wall. These nonzero adsorption and desorption times lead to an additional band broadening when conducting chromatographic separations. Shear-driven flows, requiring a noncoated moving wall and a stationary coated wall, intrinsically turn out to be more prone to this effect than pressure-driven or electro-driven flows for example. The present study also shows that, interestingly, the number of analyte-wall collisions increases with the inverse of the channel depth and not with its second power, as would be expected from the Einstein–Smoluchowski relationship for molecular diffusion.     

The apolar bilinear form in geometric modeling

Some recent methods of Computer Aided Geometric Design are related to the apolar bilinear form, an inner product on the space of homogeneous multivariate polynomials of a fixed degree, already known in 19th century invariant theory. Using a generalized version of this inner product, we derive in a straightforward way some of the recent results in CAGD, like Marsden's identity, the expression for the de Boor-Fix functionals, and recursion schemes for the computation of B-patches and their derivatives.

     

Equivalence of functional limit theorems for stationary point processes and their Palm distributions

Summary Let P be the distribution of a stationary point process on the real line and let P ⁰ be its Palm distribution. In this paper we consider two types of functional limit theorems, those in terms of the number of points of the point process in (0, t] and those in terms of the location of the nth point right of the origin. The former are most easily obtained under P and the latter under P ⁰. General conditions are presented that guarantee equivalence of either type of functional limit theorem under both probability measures, and under a third, P ₁, which plays a role in the proofs and is obtained from P by shifting the origin to the first point of the process on the right.In a brief final section the obtained results for either type of functional limit theorem are extended to equivalences between the two types by applying well-known results about processes drifting to infinity and the corresponding inverse processes.     

Experimental study of the retention properties of a cyclo olefin polymer pillar array column in reversed-phase mode

Experimental measurements to study the retention capacity and band broadening under retentive conditions using micromachined non-porous pillar array columns fabricated in cyclo olefin polymer are presented. In particular, three columns with different depths but with the same pillar structure have been fabricated via hot embossing and pressure-assisted thermal bonding. Separations of a mixture of four coumarins using varying mobile phase compositions have been monitored to study the relation between the retention factor and the ratio of organic solvent in the aqueous mobile phase. Moreover, the linear relation between the retention and the surface/volume ratio predicted in theory has been observed, achieving retention factors up to k=2.5. Under the same retentive conditions, minimal reduced plate height values of h(min)=0.4 have been obtained at retention factors of k=1.2. These experimental results are compared with the case of non-porous and porous silicon pillars. Similar results for the plate heights are achieved while retention factors are higher than the non-porous silicon column and considerably smaller than the porous pillar column, given the non-porous nature of the used cyclo olefin polymer. The feasibility of using this polymer column as an alternative to the pillar array silicon columns is corroborated.     

Hydrodynamic chromatography of polystyrene microparticles in micropillar array columns

We report on the possibility to perform HDC in micropillar array columns and the potential advantages of such a system. The HDC performance of a pillar array column with pillar diameter = 5 μm and an interpillar distance of 2.5 μm has been characterized using both a low MW tracer (FITC) and differently sized polystyrene bead samples (100, 200 and 500 nm). The reduced plate height curves that were obtained for the different investigated markers all overlapped very well, and attained a minimum value of about h_min = 0.3 (reduction based on the pillar diameter), corresponding to 1.6 μm in absolute value and giving good prospects for high efficiency separations. The obtained reduced retention time values were in fair agreement with that predicted by the Di Marzio and Guttman model for a flow between flat plates, using the minimal interpillar distance as characteristic interplate distance.     

Experimental study of the depth influence on the band broadening effect in a cyclo-olefin polymer column containing an array of ordered pillars

An experimental study of a micromachined non-porous pillar array column performance under non-retentive conditions is presented. The same pillar structure has been fabricated in cyclo-olefin polymer (COP) chips with three different depths via hot embossing and pressure-assisted thermal bonding. The influence of the depth on the band broadening along with the already known contribution arising from the top and bottom cover plates has been studied. The experimental results exhibit reduced plate heights as low as 0.2, which are in agreement with the previous experimental work. Moreover, the constant values of the reduced Van Deemter expression are also in accordance with the previous studies. A more exhaustive study of the C-term band broadening is also presented, showing that comparing the space between the pillars with different open tubular rectangular channels offers a good estimation of the C-term band broadening that is obtained experimentally. These experimental results, hence, confirm that micromachined pillar array columns fabricated in COP can achieve the same performance as the ones fabricated in silicon for the presently studied pillar channel design.     

Parameters affecting the separation of intact proteins in gradient-elution reversed-phase chromatography using poly(styrene-co-divinylbenzene) monolithic capillary columns

An experimental study was performed to investigate the effects of column parameters and gradient conditions on the separation of intact proteins using styrene-based monolithic columns. The effect of flow rate on peak width was investigated at constant gradient steepness by normalizing the gradient time for the column hold-up time. When operating the column at a temperature of 60 °C a small C-term effect was observed in a flow rate range of 1–4 μL/min. However, the C-term effect on peak width is not as strong as the decrease in peak width due to increasing flow rate. The peak capacity increased according to the square root of the column length. Decreasing the macropore size of the polymer monolith while maintaining the column length constant, resulted in an increase in peak capacity. A trade-off between peak capacity and total analysis time was made for 50, 100, and 250 mm long monolithic columns and a microparticulate column packed with 5 μm porous silica particles while operating at a flow rate of 2 μL/min. The peak capacity per unit time of the 50 mm long monolithic column with small pore size was superior when the total analysis time is below 120 min, yielding a maximum peak capacity of 380. For more demanding separations the 250 mm long monolith provided the highest peak capacity in the shortest possible time frame.