On the Existence of Completely Saturated Packings and Completely Reduced Coverings

We prove the following conjecture of G. Fejes Toth, G. Kuperberg, and W.Kuperberg: every body K in either n-dimensional Euclidean or n-dimensional hyperbolic space admits a completely saturated packing and a completely reduced covering. Also we prove the following counterintuitive result: for every >0, there is a body K in hyperbolic n-space which admits a completely saturated packing with density less than but which also admits a tiling.     

Apollonian circle packings: number theory

Apollonian circle packings arise by repeatedly filling the interstices between mutually tangent circles with further tangent circles. It is possible for every circle in such a packing to have integer radius of curvature, and we call such a packing an integral Apollonian circle packing. This paper studies number-theoretic properties of the set of integer curvatures appearing in such packings. Each Descartes quadruple of four tangent circles in the packing gives an integer solution to the Descartes equation, which relates the radii of curvature of four mutually tangent circles: . Each integral Apollonian circle packing is classified by a certain root quadruple of integers that satisfies the Descartes equation, and that corresponds to a particular quadruple of circles appearing in the packing. We express the number of root quadruples with fixed minimal element −n as a class number, and give an exact formula for it. We study which integers occur in a given integer packing, and determine congruence restrictions which sometimes apply. We present evidence suggesting that the set of integer radii of curvatures that appear in an integral Apollonian circle packing has positive density, and in fact represents all sufficiently large integers not excluded by congruence conditions. Finally, we discuss asymptotic properties of the set of curvatures obtained as the packing is recursively constructed from a root quadruple.     

A study of the practical limitations of principal components analysis and the labelling of unresolved HPLC peaks

Summary The method of iterative target transformation factor analysis (ITTFA) used in conjunction with second derivative peak finding has shown to be a practical method for the peak deconvolution and reconstruction of HPLC chromatograms and spectra. The second derivative method of peak finding is acceptable for resolutions above 0.5 for peaks of similar heights. Above 0.5 resolution the labelling gives correct results where the spectra are substantially different and also when reasonably similar. Below this value the peak labelling was still accurate where the spectra were different. Solvent effects on the spectra of the compounds studied are small and do not hinder the peak labelling process. Thus small “local” libraries are feasible. Presented at the 17th International Symposium on Chromatography, September 25–30, 1988. Vienna, Austria.     

Isolation of lignosulfonate with low polydispersity index

<正>Lignosulfonate with low polydispersity index of 1.178-1.210 was isolated by gel column chromatography of Sephacryl S-100 eluted with 0.2 mol/L of NaNO_3 aqueous solution,whereas nearly monodisperse ligosulfonate fraction with polydispersity of 1.0(57 can be obtained after chromatographic separation twice.This method provides an available approach to investigate the structure and characteristics of lignosulfonate.     

Chromatographic fingerprint study on Evodia rutaecarpa (Juss.) Benth by HPLC/DAD/ESI‐MSn technique

The LC–ESI‐MSⁿ method was developed for the analysis and characterization of alkaloids in the extract of E. rutaecarpa (called Wuzhuyu in Chinese). Thirty‐six batches of Wuzhuyu from different locations of China were investigated and the common fingerprinting profile was established with a professional analytical software recommended by the State Food and Drug Administration. Fifteen chemical components of the common peaks were identified by multi‐stage MS. The effects on the chromatographic profile resulting from different collecting locations, harvesting times or storage times were studied. Hierarchical clustering analysis and principal components analysis were also performed to classify and differentiate the 36 batches of the samples. As a result, those which had same chemical properties were sorted into one cluster, which was very useful in evaluating and controlling the quality of Wuzhuyu.     

Characterization of Ascentis RP-Amide column: Lipophilicity measurement and linear solvation energy relationships

This study investigates lipophilicity determination by chromatographic measurements using the polar embedded Ascentis RP-Amide stationary phase. As a new generation of amide-functionalized silica stationary phase, the Ascentis RP-Amide column is evaluated as a possible substitution to the n  -octanol/water partitioning system for lipophilicity measurements. For this evaluation, extrapolated retention factors, log k^′_w$\text{log}{k^{\prime }}_w$, of a set of diverse compounds were determined using different methanol contents in the mobile phase. The use of n-octanol enriched mobile phase enhances the relationship between the slope (S  ) of the extrapolation lines and the extrapolated log k^′_w$\text{log}{k^{\prime }}_w$ (the intercept of the extrapolation), as well as the correlation between log P   values and the extrapolated log k^′_w$\text{log}{k^{\prime }}_w$ (1:1 correlation, r² = 0.966). In addition, the use of isocratic retention factors, at 40% methanol in the mobile phase, provides a rapid tool for lipophilicity determination. The intermolecular interactions that contribute to the retention process in the Ascentis RP-Amide phase are characterized using the solvation parameter model of Abraham. The LSER system constants for the column are very similar to the LSER constants of the n-octanol/water extraction system. Tanaka radar plots are used for quick visual comparison of the system constants of the Ascentis RP-Amide column and the n-octanol/water extraction system. The results all indicate that the Ascentis RP-Amide stationary phase can provide reliable lipophilic data.     

Chemometric quality control of chromatographic purity

It is common practice in chromatographic purity analysis of pharmaceutical manufacturing processes to assess the quality of peak integration combined by visual investigation of the chromatogram. This traditional method of visual chromatographic comparison is simple, but is very subjective, laborious and seldom very quantitative. For high-purity drugs it would be particularly difficult to detect the occurrence of an unknown impurity co-eluting with the target compound, which is present in excess compared to any impurity. We hypothesize that this can be achieved through Multivariate Statistical Process Control (MSPC) based on principal component analysis (PCA) modeling. In order to obtain the lowest detection limit, different chromatographic data preprocessing methods such as time alignment, baseline correction and scaling are applied. Historical high performance liquid chromatography (HPLC) chromatograms from a biopharmaceutical in-process analysis are used to build a normal operation condition (NOC) PCA model. Chromatograms added simulated 0.1% impurities with varied resolutions are exposed to the NOC model and monitored with MSPC charts. This study demonstrates that MSPC based on PCA applied on chromatographic purity analysis is a powerful tool for monitoring subtle changes in the chromatographic pattern, providing clear diagnostics of subtly deviating chromatograms. The procedure described in this study can be implemented and operated as the HPLC analysis runs according to the process analytical technology (PAT) concept aiming for real-time release.     

Determination of antibiotic residues in honey

Honey produced by honeybees is a valuable food product. The presence of xenobiotics in honey may harm its quality and constitute a danger to human health. Antibiotics are commonly applied by beekeepers to eliminate disease among honeybees. Moreover, ubiquitous administration of antibiotics may cause bacteria to become resistant to many drugs and spread antibiotic-resistant strains of bacteria. Appropriate sample preparation and determination of antibiotics at very low concentrations in foodstuffs are real analytical challenges. This article reviews analytical methods used for determination of residues of different sorts of antibiotic in honey and other honeybee products.     

Method development for the determination of 24S‐hydroxycholesterol in human plasma without derivatization by high‐performance liquid chromatography with tandem mass spectrometry in atmospheric pressure chemical ionization mode

We developed a highly sensitive and specific high‐performance liquid chromatography with tandem mass spectrometry method with an atmospheric pressure chemical ionization interface to determine 24S‐hydroxycholesterol, a major metabolite of cholesterol formed by cytochrome P450 family 46A1, in human plasma without any derivatization step. Phosphate buffered saline including 1% Tween 80 was used as the surrogate matrix for preparation of calibration curves and quality control samples. The saponification process to convert esterified 24S‐hydroxycholesterol to free sterols was optimized, followed by liquid–liquid extraction using hexane. Chromatographic separation of 24S‐hydroxycholesterol from other isobaric endogenous oxysterols was successfully achieved with gradient mobile phase comprised of 0.1% propionic acid and acetonitrile using L‐column2 ODS (2 μm, 2.1 mm id × 150 mm). This assay was capable of determining 24S‐hydroxycholesterol in human plasma (200 μL) ranging from 1 to 100 ng/mL with acceptable intra‐ and inter‐day precision and accuracy. The potential risk of in vitro formation of 24S‐hydroxycholesterol by oxidation from endogenous cholesterol in human plasma was found to be negligible. The stability of 24S‐hydroxycholesterol in relevant solvents and human plasma was confirmed. This method was successfully applied to quantify the plasma concentrations of 24S‐hydroxycholesterol in male and female volunteers.     

新解色谱现象

在客观分析色谱现象的基础上对色谱分离机理的传统解释提出了质疑和否定,并对色谱分离机理作出了新的解释。通过对色谱现象的研究和分析,不仅确定了一条流体运动的基本规律,还对固定相赋予了新的内涵,提出了能量场固定相的设想。由于对色谱分离原理进行了新的解释,因此对色谱现象提出了一个新的定义。