How should one define a (weak) crystal?

We compare two proposals for the study of positional long-range order: one in terms of the spectrum of the translation operator, the other in terms of the Fourier spectrum. We point out that only the first one allows for the consideration of molecular, as opposed to atomic, (weakly) periodic structures. We illustrate this point on the Thue-Morse system.     

Avoider-Enforcer games

Let p and q be positive integers and let H be any hypergraph. In a (p,q,H) Avoider-Enforcer game two players, called Avoider and Enforcer, take turns selecting previously unclaimed vertices of H. Avoider selects p vertices per move and Enforcer selects q vertices per move. Avoider loses if he claims all the vertices of some hyperedge of H; otherwise Enforcer loses. We prove a sufficient condition for Avoider to win the (p,q,H) game. We then use this condition to show that Enforcer can win the (1,q) perfect matching game on K2n for every q?cn/logn for an appropriate constant c, and the (1,q) Hamilton cycle game on Kn for every q?cnloglogloglogn/lognlogloglogn for an appropriate constant c. We also determine exactly those values of q for which Enforcer can win the (1,q) connectivity game on Kn. This result is quite surprising as it substantially differs from its Maker-Breaker analog. Our method extends easily to improve a result of Lu [X. Lu, A note on biased and non-biased games, Discrete Appl. Math. 60 (1995) 285-291], regarding forcing an opponent to pack many pairwise edge disjoint spanning trees in his graph.     

HPLC-separation of cis and trans monounsaturated fatty acids

Summary The chromatography of monounsaturated fatty acids as their methyl esters on silver nitrate-loaded HPLC-columns has been studied. The separation of cis- and trans-isomers was easily achieved even with columns of low performance. High-performance small-particle-columns treated with silver nitrate separated a large variety of monounsaturated cis and trans positional isomers. The influence of variable silver-loads on the selectivity of the system was studied and a survey of the distribution of positional trans C₁₈₁ isomers in commercial margarines is given.     

Resolution of triacylglycerol positional isomers by reversed-phase high-performance liquid chromatography

The ability of reversed-phase high-performance liquid chromatography (RP-HPLC) to separate some positionally isomeric disaturated and monounsaturated triacylglycerols (TAGs) as intact species is demonstrated for the first time. Mobile phases of acetonitrile modified with methanol, ethanol, 2-propanol, 1-propanol, 1-butanol, acetone, or dichloromethane were tested for the separation of POP-PPO, PLP-PPL, PEP-PPE, and PDP-PPD (P-palmitic, O-oleic, L-linoleic, E-eicosapentaenoic, D-docosahexaenoic acid residue) on a single RP-HPLC column. The resolution improved with increasing number of double bonds in the acyl residues. While POP and PPO were only partially resolved, PDP and PPD were fully separated with all tested mobile phases, except those containing methanol. Also separated were the four TAGs having the same equivalent carbon number (ECN = 42), PEP, PPE, PDP, and PPD, on a single RP-HPLC column with mobile phase acetonitrile-2-propanol (70:30, v/v) at 0.8 mL/min. In all cases the isomer with the unsaturated acyl residue in either 1- or 3-position was retained more strongly than the respective 2-isomer.     

Simple matching vs linear assignment in scheduling models with positional effects: A critical review

This paper addresses scheduling models in which a contribution of an individual job to the objective function is represented by the product of its processing time and a certain positional weight. We review most of the known results in the area and demonstrate that a linear assignment algorithm as part of previously known solution procedures can be replaced by a faster matching algorithm that minimizes a linear form over permutations. Our approach reduces the running time of the resulting algorithms by up to two orders, and carries over to a wider range of models, with more general positional effects. Besides, the same approach works for the models with no prior history of study, e.g., parallel machine scheduling with deterioration and maintenance to minimize total flow time.     

Integrated multidimensional and comprehensive 2D GC analysis of fatty acid methyl esters

Fatty acid methyl ester (FAME) profiling in complex fish oil and milk fat samples was studied using integrated comprehensive 2D GC (GC × GC) and multidimensional GC (MDGC). Using GC × GC, FAME compounds – cis‐ and trans‐isomers, and essential fatty acid isomers – ranging from C18 to C22 in fish oil and C18 in milk fat were clearly displayed in contour plot format according to structural properties and patterns, further identified based on authentic standards. Incompletely resolved regions were subjected to MDGC, with Cn (n = 18, 20) zones transferred to a ²D column. Elution behavior of C18 FAME on various ²D column phases (ionic liquids IL111, IL100, IL76, and modified PEG) was evaluated. Individual isolated Cn zones demonstrated about four‐fold increased peak capacities. The IL100 provided superior separation, good peak shape, and utilization of elution space. For milk fat‐derived FAME, the ²D chromatogram revealed at least three peaks corresponding to C18:1, more than six peaks for cis/trans‐C18:2 isomers, and two peaks for C18:3. More than 17 peaks were obtained for the C20 region of fish oil‐derived FAMEs using MDGC, compared with ten peaks using GC × GC. The MDGC strategy is useful for improved FAME isomer separation and confirmation.     

自制环糊精衍生物毛细管柱气相色谱法分离氯苯甲醛和苯系物位置异构体

研究了自制的4根不同环糊精毛细管色谱柱对氯苯甲醛异构体的分离特性,结果表明环糊精类毛细管柱对氯苯甲醛异构体有特殊的分离性能,其中硅烷化乙酰化环糊精柱对其分离效果最佳。此外,全甲基环糊精色谱柱对酚类体系显示了很好的分离性能,硅烷化乙酰化环糊精柱对卤代甲苯体系也取得了令人满意的分离效果。     

Economical tight examples for the biased Erd?s-Selfridge theorem

A positional game is essentially a generalization of tic-tac-toe played on a hypergraph (V,F). A pivotal result in the study of positional games is the Erd?s-Selfridge theorem, which gives simple criteria for the existence of a Breaker's winning strategy on a hypergraph F. It has been shown that the Erd?s-Selfridge theorem can be tight and that numerous extremal systems exist for that theorem. We focus on a generalization of the Erd?s-Selfridge theorem proven by Beck for biased (p:q) games, which we call the (p:q)-Erd?s-Selfridge theorem. We show that for pn-uniform hypergraphs there is a unique extremal system for the (p:q)-Erd?s-Selfridge theorem (q?2) when Maker must win in exactly n turns (i.e., as quickly as possible).     

Exploring antibody polyspecificity using synthetic combinatorial libraries

Summary Extensive mapping studies for seven antigen-antibody interactions have been carried out using both individual analogs and peptide libraries. With competitive ELISA, these studies have revealed that monoclonal antibodies exhibit a broad range of specificities, from antibodies that recognize only conservative substitutions for 1–2 positions of the antigenic determinant, to antibodies that recognize sequences that are completely unrelated to the parent antigen with comparable affinities. Synthetic combinatorial libraries, containing millions of peptide sequences, permit a more systematic and rapid evaluation of the extent of multiple-binding specificities of monoclonal antibodies than individual analogs. The peptide libraries used here comprise mixtures of compounds having specifically defined positions and mixture positions. The same diversity of sequences in different formats, which differ by the numbers of positions singularly defined and different locations defined within the sequence, can be examined. Comparison of the screening results, selection criteria of the most active mixtures, and different approaches used for the deconvolution of active individual compounds are discussed. Synthetic combinatorial libraries greatly facilitate the understanding of antigen-antibody interactions at the amino acid level and will assist in the development of improved immunodiagnostics.     

Regioselective separation of isomeric triacylglycerols by reversed-phase high-performance liquid chromatography: stationary phase and mobile phase effects

Complete regioselective separation of five pairs of isomeric dipalmitoyl polyalkenoyl glycerols with two to six double bonds in the unsaturated acyl residues has been achieved by RP-HPLC on a single ODS column. Four ODS columns with stationary phases containing different percentages of free silanol groups have been tested. Binary mobile phases of ACN admixed with dichloromethane, tetrahydrofuran, 1-propanol, 2-propanol, ethanol, or acetone have been examined. The choice of modifier depended on the nature of the stationary phase. The more polar solvents were better suited for stationary phases with higher percentage of free silanol groups. Isomeric species were eluted according to chain length, number of double bonds, and the position of the unsaturated acyl chain in the glycerol molecule. Retention increases in the order 20:5 < 22:6 < 18:3 < 20:4 < 18:2. Within each isomeric pair, the species with unsaturated acyl chain occupying either the sn-1- or the 3-position were retained preferentially. Complete simultaneous regioselective separation of 10 isomeric triacylglycerols in a single isocratic run on a single ODS column was demonstrated.